IntroductionA centerpiece of representation theory is the Schur-Weyl duality, which says that, (a) the general linear group GL n (C) and the symmetric group S k both act on tensor spacewith dim(V ) = n, (b) these two actions commute and (c) each action generates the full centralizer of the other, so that (d) as a (GL n (C), S k )-bimodule, the tensor space has a multiplicity free decomposition,

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

We introduce an associative algebra M k (x) whose dimension is the 2k-th Motzkin number. The algebra M k (x) has a basis of "Motzkin diagrams," which are analogous to Brauer and Temperley-Lieb diagrams. We show for a particular value of x that the algebra M k (x) is the centralizer algebra of the quantum enveloping algebra U q (gl 2 ) acting on the k-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible U q (gl 2 )-modules. We prove that M k (x) is cellular in the sense of Graham and Lehrer and construct indecomposable M k (x)-modules which are the left cell modules. When M k (x) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible M k (x)-modules. We compute the determinant of the Gram matrix of a bilinear form on the cell modules and use these determinants to show that M k (x) is semisimple exactly when x is not the root of certain Chebyshev polynomials.

The q-rook monoid I n (q) is a semisimple algebra over C(q) that specializes when q → 1 to C[R n ], where R n is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. When q is specialized to a prime power,is the monoid of n × n matrices with entries from a finite field having q-elements and B ⊆ M is the Borel subgroup of invertible upper triangular matrices. In this paper, we (i) give a new presentation for I n (q) on generators and relations and determine a set of standard words which form a basis; (ii) explicitly construct a complete set of "seminormal" irreducible representations of I n (q); and (iii) show that I n (q) is the centralizer of the quantum general linear group U q gl(r) acting on the tensor product (W ⊕ V ) ⊗n , where V is the fundamental U q gl(r) module and W is the trivial U q gl(r) module.  2004 Elsevier Inc. All rights reserved.

In honor of Richard Stanley on his 60th birthday. AbstractWe give combinatorial proofs of two identities from the representation theory of the partition algebra CA k (n), n ≥ 2k. The first is n k = λ f λ m λ k , where the sum is over partitions λ of n, f λ is the number of standard tableaux of shape λ, and m λ k is the number of "vacillating tableaux" of shape λ and length 2k. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is B(2k) = λ (m λ k ) 2 , where B(2k) is the number of set partitions of {1, . . . , 2k}. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.